Tag Archives: strategies

Just the Facts Again

A month ago I wrote a post called Just the Facts where I talked about the way my daughter has been practicing her multiplication facts with me at home. In case you don’t want to check out that post, here’s a quick recap of the way she’s practicing her facts:

  1. She answers as many multiplication flash cards as she can in one minute. She does two trials of one minute each.
  2. She counts the cards after each trial.
  3. I monitor and provide feedback as needed.
  4. After two trials, she graphs her higher score.

Here’s how she was doing after five sessions back in late August:

And here’s how she’s been doing after 16 total sessions between August and now:

I feel like I’m supposed to do some statistical analysis of this and talk about the median or mode or something, but I don’t really care to be so formal. These are our informal takeaways:

  • She has good days and better days. We don’t focus too much on her results for any given day.
  • She appreciates getting two trials each time because if she blows it on one trial she knows she can make up for it on the other trial.
  • She and I were both excited when she set a new record of 23 correct in one minute.

I’m so proud to see her flexibly use appropriate strategies for finding various products. She can double like no one’s business when she sees a factor of 2, 4, or 8, but she can just as quickly build up or down from a 5 or 10s fact when she sees a factor of 6 or 9.

I wrote earlier about how she struggles with mentally doubling numbers like 18 and 36. I offered to let her use a whiteboard to help her with that doubling when she needs it, and that has been a huge help. I’m especially proud that it hasn’t turned into a crutch for every problem. She really only uses it when she knows the number crunching in her head is too much for her to handle.

For a few sessions, I was doing some flash card practice with her around those challenging doubles, but I decided to move away from it. After I wrote my previous post, Michael Pershan shared this excellent post called What People Get Wrong About Memorizing Math Facts. He said something in the post that I needed reminding of:

“…the best practice for remembering something is practicing remembering it.”

This led me to change tactics. I created a set of flash cards of facts that my daughter is able to solve using a strategy, but I want to give her an opportunity to practice remembering the products.

When we practice these facts, it is untimed, though I only give her about 5 seconds per card, otherwise I can see her start using a strategy to derive the answer. I usually run through these cards once or twice before her two trials for that session. I told her the goal here is to practice remembering the products. I don’t want her to try and use a strategy. I just want her to see if her brain can pull up the answer from memory. I told her it’s fine if it can’t, but the act of trying to remember is a good thing to practice. I also made clear that during her two one-minute trials I still want her to try to remember these products, but if she doesn’t it’s totally okay to use a strategy at that point.

We’ve been using these cards for a while now, and I’m noticing that slowly she is starting to remember some of the products, though not all of them. What I’m really excited to see is that if she doesn’t quite remember the product I’ll ask, “What tens do you think the product is in?” and she is getting pretty good about knowing which ones are in the 50s or 60s even if she doesn’t remember if the exact product is 54 or 56.

So that’s where we are now. We only do this facts practice 1-2 times per week now that school has started. Here’s what I like about it:

  • It’s quick to do.
  • When we first started, she felt pretty down on the days when she only got 10-14 correct. But because we’ve continued doing it and graphing her results, she sees that those days are blips in an overall pattern of success.
  • She loves that there are two trials each time we do it. That feeling of getting a second chance is powerful.
  • I’ve been able to watch her work and provide support and modifications that specifically help her be successful and feel confident.
  • We’re able to work on dual goals of memorization and strategy use.

My next step is going to be introducing division flash cards into the mix. We’ve done some work relating multiplication and division already, and we’ve specifically talked about how thinking of a related multiplication fact can help her solve a division fact. I expect some bumps as that gets started, but I feel confident she’s on a good path.

Just the Facts

In my previous post (Link), I shared how I’ve recently starting doing math with my daughter to help her get warmed up for the start of 4th grade. In that post I talked about how I’m using the centers from the Illustrative Mathematics K-5 curriculum (Link) to revisit and practice working with multiplication and arrays.

In the six and half years I worked as a district math curriculum coordinator, a common concern I heard from 4th and 5th grade teachers is that their students don’t come in knowing their multiplication facts. I can attest that my daughter learned a lot about multiplication and division in 3rd grade, but I’ll be honest, she hasn’t done a whole lot of multiplying or dividing this summer (not to mention fluency is something that tends to develop over a period of years, not months). It comes as absolutely no surprise to me that she’s rusty, particularly with knowing her multiplication facts. I’m going to go out on a limb and claim that a lot of kids are rusty at the start of a new school year. We need to give them grace, which means not saying things like, “Didn’t your teacher teach this last year?” We also need to intentionally build in opportunities to practice and dust off the mental cobwebs.

Today I’d like to share how my daughter and I have been practicing multiplication facts. What I like about what we’re doing is that (1) it only takes a few minutes a day, (2) it reinforces flexible use of strategies, and (3) it gives her a second chance everyday. I got this idea from a free math intervention called Pirate Math Equation Quest (Link), developed by Dr. Katherine Berry and Dr. Sarah Powell from the Meadows Center (Link) at The University of Texas at Austin. Their intervention includes a component called Math Fact Flaschards that goes like this:

  • Student completes two trials of Math Fact Flashcards, each for 1 minute
  • Teacher and student count cards after each timing
  • Teacher monitors and provides feedback as needed
  • After 2 trials, student graphs the higher score

Rather than use traditional flashcards, I created flashcards that show two facts per card, the initial fact and its turnaround. For example, the card with 2 × 5 also shows 5 × 2. I got this idea from the 4th grade Investigations 2nd edition curriculum. It reinforces the idea that every time you know the answer for one fact, you really know the answer for two (with the exception of square numbers).

Four multiplication flash cards.

Before we start a trial, I always remind her that she is going to “just know” some of the facts because she’s so familiar with them, but for the ones she doesn’t “just know” she can use one of the multiplication strategies she’s learned. The following poster is hanging on the wall next to where she’s sitting so she can turn and reference it as needed.

Table listing the multiplication thinking strategies

These are the thinking strategies developed by Origo Education (Link). If you’re not familiar with them, check out this YouTube playlist that includes one-minute videos explaining each strategy. (Link) If you want to see how a child uses one of the strategies, here’s a link to a short video of my daughter talking through the Build Down strategy she used to solve 9 × 7. (Link)

Please note, you can’t just throw strategies at your students. They have to be intentionally introduced and practiced, but it is well worth the time! Students who lack a robust toolbox of strategies have to rely solely on memorization (which is a big ask!) or inefficient strategies like skip counting. If you’re interested in learning more about how to teach these strategies, Origo has a great series called The Book of Facts that shares activities and games for teaching a set of fact strategies for each of the four operations. (Link)

During each trial, I present the flashcards one at a time. I put all of the ones she answers correctly in a pile and any she answers incorrectly in another pile. After the minute is over, she counts the number correct, and then we discuss the ones she answered incorrectly. Sometimes her incorrect answers are because of a simple mistake, and I reinforce that it’s fine because she has been able to recognize the error herself. However, sometimes it’s more than a simple error. I was able to pick up very quickly that she’s also rusty with doubling 2-digit numbers that involve bridging a ten. For example, to solve 4 × 7, she can easily double 7 to get 14 and double 14 to get 28. However, to solve 4 × 8, she can easily double 8 to get 16 but she gets stuck doubling 16. Her answer might be 26 or 36.

Based on this observation, I’ve added in practice with doubling 2-digit numbers. This practice is untimed for now, though I might eventually add these cards into the deck of multiplication flashcards.

Set of flashcards that say things like "Double 25" and "Double 18"

At the end of the two trials, we graph her higher score for the day. I really love this because if she blows the first trial for whatever reason, she knows she’s going to get a second chance to get a higher score. It really takes the pressure off.

We’ve only been doing it for a week, so there’s not a lot of data to look at, but I’ve already used her graph to talk about how we all have good days and better days. I also reinforce that while some days are lower, her rate of incorrect responses is consistently low. She only ever misses 0, 1, or rarely 2 cards during a trial. She’s also been really good about stopping and thinking of an appropriate strategy whenever she gets stuck, and she is doing a great job of executing her chosen strategy accurately.

Bar graph showing the number of multiplication flash cards answered correctly each day. Bars range in height from 14 to 21.

For full transparency, her deck of flashcards includes all of the facts including the “easy” ones like 0s and 1s facts, and I’m okay with that. They’re still facts and she needs to know them. The important thing is that I continue to monitor to uncover any issues where I can support her, like with doubling 2-digit numbers. Eventually I might ween the deck down to the ones that need more intensive practice.

I like that this practice doesn’t take a lot of time, only about 3-5 minutes. If you’d like to try this out in your classroom, you might consider doing it in small groups, which is an idea shared in the Pirate Math Equation Quest intervention I mentioned earlier. During the one-minute trial, the teacher goes around the group round robin style, showing one flashcard to each student. All of the flashcards are placed in one pile and the total correct is the group’s score. The goal as a group is to try to get more and more correct each time. I like that this allows for a bit of a tradeoff. The teacher doesn’t have to feel pressured to run this activity individually with every student, but at the same time, she can learn something about each student as she conducts these trials in small groups. I’m doing this with my daughter everyday, but a teacher might be able to make small groups such that she ends up seeing every student every 3-4 days.

As I was reading over the small group directions, I realized they recommend letting the student continue trying until they get the answer correct. If the student answers incorrectly, the teacher intervenes with a suggestion such as a strategy a student might use. I think I might try that with my daughter rather than setting aside incorrect answers. Helping in the moment seems much more powerful than helping at the end. It also does a better job of validating the power of identifying and correcting mistakes. I like forward to seeing how it goes next week!